On strong separations from AC°

Abstract

We investigate sets that are immune to AC0; that is, sets with no infinite subset in AC0. First we show that such sets exist in Ppp. Although this seems like a rather weak result (since AC0 is an extremely weak complexity class and Ppp contains the entire polynomial hierarchy) we also prove a somewhat surprising theorem, showing that a significant breakthrough will be necessary in order to prove a bound much better than Ppp. Namely, we show that any answer to the question: Are there sets in NP that are immune to AC0? will provide non-relativizable proof techniques suitable for attacking the Ntime vs Dtime question. We also show the related result that ACC is properly contained in PP, and that there is no uniform family of ACC circuits that computes the permanent of two matrices. This seems to be the first example of a lower bound in circuit complexity where the uniformity condition is essential; it is still unknown if there is any set in Dtime(2n0(1)) that does not have non-uniform ACC circuits.